Add plot for characteristic curves in rank 0 case

main.tex

@@ -282,9 +282,7 @@ Considering the stability conditions with two parameters $\alpha, \beta$ on
 Picard rank 1 surfaces.
 We can draw 2 characteristic curves for any given Chern character $v$ with
 $\Delta(v) \geq 0$ and positive rank.
-These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$, and
-are illustrated in Fig \ref{fig:charact_curves_vis}
-(dotted line for $i=1$, solid for $i=2$).
+These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
 
 \begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
 Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
@@ -296,9 +294,21 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
 \end{align*}
 \end{definition}
 
-\begin{fact}[Geometry of Characteristic Curves]
+\subsection{Geometry of the Characteristic Curves}
+
+These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
+not affected by flipping the sign of $v$ so it's only necessary to consider
+non-negative rank.
+As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
+has Gieseker stable coherent sheaves appearing in the heart of the stability
+condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
+
+\subsubsection{Positive Rank Case}
+\label{subsect:positive-rank-case-charact-curves}
+
+\begin{fact}[Geometry of Characteristic Curves in Positive Rank Case]
 The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
-as well as the restrictions on $v$:
+as well as the restrictions on $v$, when $\chern_0(v)>0$:
 \begin{itemize}
 	\item $V_v$ is a vertical line at $\beta=\mu(v)$
 	\item $\Theta_v$ is a hyperbola with assymptotes angled at $\pm 45^\circ$
@@ -312,21 +322,8 @@ as well as the restrictions on $v$:
 \end{itemize}
 \end{fact}
 
-
-\subsection{Relevance of the Vertical Line $V_v$}
-
-By definition of the first tilt $\firsttilt\beta$, objects of Chern character
-$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
-$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
-in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
-fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
-$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
-$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
-Because of this, when using these characteristic curves, only positive ranks are
-considered, as negative rank objects are implicitly considered on the right hand
-side of $V_v$.
-
-
+These are illustrated in Fig \ref{fig:charact_curves_vis}
+(dotted line for $i=1$, solid for $i=2$).
 
 \begin{sagesilent}
 from characteristic_curves import \
@@ -334,6 +331,7 @@ typical_characteristic_curves, \
 degenerate_characteristic_curves
 \end{sagesilent}
 
+
 \begin{figure}
 \centering
 \begin{subfigure}{.49\textwidth}
@@ -358,8 +356,49 @@ degenerate_characteristic_curves
 \label{fig:charact_curves_vis}
 \end{figure}
 
+\begin{sagesilent}
+from rank_zero_case import Theta_v_plot
+\end{sagesilent}
+
+\subsubsection{Rank Zero Case}
+\begin{fact}[Geometry of Characteristic Curves in Rank 0 Case]
+The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
+as well as the restrictions on $v$, when $\chern_0(v)=0$ and $\chern_1(v)>0$:
+
+
+\begin{minipage}{0.49\textwidth}
+\begin{itemize}
+	\item $V_v = \emptyset$
+	\item $\Theta_v$ is a vertical line at $\beta=\frac{D}{C}$
+		where $v=\left(0,C\ell,D\ell^2\right)$
+\end{itemize}
+\end{minipage}
+\begin{minipage}{0.49\textwidth}
+	\sageplot[width=\textwidth]{Theta_v_plot}
+	%\caption{$\Delta(v)>0$}
+	%\label{fig:charact_curves_rank0}
+\end{minipage}
+\end{fact}
 
-\subsection{Relevance of the Hyperbola $\Theta_v$}
+
+
+\subsection{Relevance of $V_v$}
+\label{subsect:relevance-of-V_v}
+
+By definition of the first tilt $\firsttilt\beta$, objects of Chern character
+$v$ can only be in $\firsttilt\beta$ on the left of $V_v$, where
+$\chern_1^{\alpha,\beta}(v)>0$, and objects of Chern character $-v$ can only be
+in $\firsttilt\beta$ on the right, where $\chern_1^{\alpha,\beta}(-v)>0$. In
+fact, if there is a Gieseker semistable coherent sheaf $E$ of Chern character
+$v$, then $E \in \firsttilt\beta$ if and only if $\beta<\mu(E)$ (left of the
+$V_v$), and $E[1] \in \firsttilt\beta$ if and only if $\beta\geq\mu(E)$.
+Because of this, when using these characteristic curves, only positive ranks are
+considered, as negative rank objects are implicitly considered on the right hand
+side of $V_v$.
+
+
+
+\subsection{Relevance of $\Theta_v$}
 
 Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
 $\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
@@ -377,6 +416,7 @@ $\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
 $\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
 (as per subsection \ref{subsect:bertrams-nested-walls}).
 
+
 \subsection{Bertram's Nested Wall Theorem}
 \label{subsect:bertrams-nested-walls}